CppAD swMATH ID: 4866 Software Authors: Bell, Bradley M.; Burke, James V. Description: Algorithm differentiation of implicit functions and optimal values In applied optimization, an understanding of the sensitivity of the optimal value to changes in structural parameters is often essential. Applications include parametric optimization, saddle point problems, Benders decompositions, and multilevel optimization. In this paper we adapt a known automatic differentiation (AD) technique for obtaining derivatives of implicitly defined functions for application to optimal value functions. The formulation we develop is well suited to the evaluation of first and second derivatives of optimal values. The result is a method that yields large savings in time and memory. The savings are demonstrated by a Benders decomposition example using both the ADOL-C and CppAD packages. Some of the source code for these comparisons is included to aid testing with other hardware and compilers, other AD packages, as well as future versions of ADOL-C and CppAD. The source code also serves as an aid in the implementation of the method for actual applications. In addition, it demonstrates how multiple C++ operator overloading AD packages can be used with the same source code. This provides motivation for the coding numerical routines where the floating point type is a C++ template parameter. Homepage: http://www.coin-or.org/CppAD/ Keywords: automatic differentiation; Newton’s method; iterative process; implicit function; parametric programming; C++ template functions; ADOL-C; CppAD Related Software: Ipopt; ADOL-C; SCIP; MINLPLib; CPLEX; Eigen; R; AMPL; TAPENADE; FADBAD++; LINDO; Bonmin; COIN-OR; Stan; ADIC; TMB; HSL; FEASPUMP; LINDOGlobal; MINLP Cited in: 34 Documents all top 5 Cited by 71 Authors 8 Gleixner, Ambros M. 5 Müller, Benjamin 4 Berthold, Timo 3 Serrano, Felipe 2 Andersson, Joel A. E. 2 Bell, Bradley M. 2 Diehl, Moritz Mathias 2 Gasse, Maxime 2 Lebiedz, Dirk 2 Lodi, Andrea 2 Muñoz, Gonzalo 2 Siehr, Jochen 2 van Leeuwen, Tristan 2 Weltge, Stefan 1 Åkesson, Johan 1 Aravkin, Aleksandr Y. 1 Askham, Travis 1 Auer, Ekaterina 1 Batenburg, Kees Joost 1 Baydin, Atılım Güneş 1 Bessac, Julie 1 Bischof, Christian H. 1 Błaszczyk, Jacek 1 Bonat, Wagner Hugo 1 Burke, James V. 1 Chauris, Hervé 1 Cocher, Emmanuel 1 Delinchant, Benoit 1 Dumitrel, Vlad 1 Enciu, P. 1 Flaxman, Abraham D. 1 Fourer, Robert 1 Gerbaud, Laurent 1 Gillis, Joris J. M. 1 Held, Harald 1 Hogan, Robin J. 1 Horn, Greg 1 Hovland, Paul D. 1 Huang, Wei 1 Jones, Colin Neil 1 Kronqvist, Jan 1 Kulshreshtha, Kshitij 1 Kutz, J. Nathan 1 Laureano, Henrique Aparecido 1 Listov, Petr 1 Lundell, Andreas 1 Luther, Wolfram Jérôme 1 Ma, Jun 1 MacIntyre, K. 1 Maher, Stephen J. 1 Maretzke, Simon 1 Martin, Richard Kipp 1 Narayanan, Sri Hari Krishna 1 Norris, Boyana 1 Parreira da Silva, Guilherme 1 Pearlmutter, Barak A. 1 Pedroso, João Pedro 1 Petterle, Ricardo Rasmussen 1 Plessix, René-Edouard 1 Radul, Alexey Andreyevich 1 Rawlings, James B. 1 Ribeiro, Paulo Justiniano jun. 1 Sielemann, M. 1 Siskind, Jeffrey Mark 1 Skanda, Dominik 1 Sluşanschi, Emil I. 1 Szynkiewicz, Wojciech 1 Unger, Jonas 1 Vigerske, Stefan 1 Westerlund, Tapio 1 Wurtz, F. all top 5 Cited in 20 Serials 3 Journal of Global Optimization 3 Mathematical Programming. Series A. Series B 2 Inverse Problems 2 ACM Transactions on Mathematical Software 2 SIAM Journal on Scientific Computing 2 International Journal of Applied Mathematics and Computer Science 2 Mathematical Programming Computation 1 Mathematics and Computers in Simulation 1 Optimal Control Applications & Methods 1 COMPEL 1 Annals of Operations Research 1 Journal of Statistical Computation and Simulation 1 SIAM Journal on Optimization 1 Computational Optimization and Applications 1 Optimization Methods & Software 1 Higher-Order and Symbolic Computation 1 Computational Geosciences 1 Journal of Machine Learning Research (JMLR) 1 SIAM Journal on Applied Dynamical Systems 1 Numerical Algebra, Control and Optimization all top 5 Cited in 13 Fields 20 Operations research, mathematical programming (90-XX) 15 Numerical analysis (65-XX) 7 Computer science (68-XX) 4 Systems theory; control (93-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Statistics (62-XX) 2 Classical thermodynamics, heat transfer (80-XX) 1 General and overarching topics; collections (00-XX) 1 Mechanics of particles and systems (70-XX) 1 Optics, electromagnetic theory (78-XX) 1 Geophysics (86-XX) 1 Biology and other natural sciences (92-XX) Citations by Year